Trailing edge noise produced by the scattering of boundary layer turbulence

Noise generated at the trailing edges of airfoils can be an important contributor to the sound levels of aircraft and turbomachinery. Previous work on trailing edge noise has largely neglected the role of the mean boundary-layer profile in the sound generation process. In the present work, a trailing edge scattering problem is formulated using a piecewise-linear symmetric mean flow with a non-zero slip velocity at the centerline, providing a representation that is intermediate between the boundary layer and wake flows. The airfoil is modeled as a zero-thickness, semi-infinite flat plate, appropriate for noise generation by sharp trailing edges in the presence of a high Reynolds number flow. The mean boundary-layer profile adjacent to the airfoil surface supports naturally-occurring vortical disturbances that have a non-trivial pressure field and convect at speeds between the slip and free stream velocities. Using these pressure disturbances to represent the turbulent field that is incident on the trailing edge, a mixed boundary value problem is formulated and solved using the Wiener-Hopf technique. This scattering problem is solved for both low Mach number and O(1) Mach number flows. The results show that the presence of the mean flow profile can significantly increase the amplitude of the sound radiated to the far field. In the O(1) Mach number case, the directivity of the scattered field is also appreciably altered. The results of the scattering problem are used along with a simplified model of the wall pressure wavenumber-frequency spectrum to generate a prediction for the power spectrum of the scattered sound field.